Investigation of weakly-nonlinear development of unsteady Görtler vortices

We have examined, both experimentally (using fully controlled disturbances) and theoretically, the weakly-nonlinear development stages of unsteady (in general) Gortler instability of a boundary layer over a concave surface. Primary attention was given to early manifestations of nonlinearity in the development of unsteady Gortler vortices belonging to the first, most rapidly growing, mode in the discrete spectrum of the stability problem. We have investigated the manifestations of instability versus the frequency of the fundamental (primary) Gortler mode and the initial disturbance amplitude. The weakly-nonlinear stage of development of unsteady Gortler vortices was found to display the following characteristic features: (a) nonlinear interaction among the combination modes in the frequency-wavenumber spectrum, (b) distortion of the wall-normal profiles of disturbance amplitudes and phases, (c) reduction of the growth rate of the fundamental Gortler mode and the majority of combination modes, and (d) a decrease in the phase velocities of unsteady disturbances. It was found that the disturbances enter the region of weakly-nonlinear development after the amplitude of the fundamental frequency-wavenumber mode reaches a threshold of 4–6 %, this value being much greater than that for Tollmien — Schlichting waves (1–2 %) but significantly lower than that for the cross-flow instability modes in three-dimensional boundary layer (more than 10 %).